Some central concepts used in the following text are briefly defined here.
Laterally Propagating Plate Wave Modes
Bulk acoustic thickness vibration arising in a piezoelectric plate (a wafer or a thin film layer) can propagate laterally (in the horizontal direction) in the plate. Such laterally propagating wave modes are called plate wave modes or Lamb wave modes [1]. The term plate wave mode will be used here. The wave propagates with a lateral wavelength λx and has a lateral wave number kx.
Different types of thickness vibration can propagate as plate waves. Examples of such vibration types are the thickness-extensional (TE) and thickness-shear (TS) vibration (FIG. 1). In the former, the particle displacement is in the thickness direction of the piezoelectric plate (z-axis in FIG. 1), and in the latter, it is perpendicular to the thickness direction. A thickness resonance arises within the piezoelectric plate when the thickness t of the plate is equal to an integer number of half-wavelengths λz: t=Nλz/2, λz=v/f, where N is an integer, v is the velocity of the acoustic wave in the piezoelectric material, and f is the operation frequency. In the first-order mode, N=1 and there is one half-wavelength accommodated within the thickness t.
In FIG. 1, some plate wave types are depicted. From top to bottom: the first-order thickness-extensional mode TE1, the second-order thickness-shear mode TS2, the first-order thickness-shear mode TS1, flexural mode. Propagation is in the lateral direction with lateral wavelength λx.
Lateral Standing Wave Resonances
Laterally propagating plate waves can be reflected from discontinuities, such as electrode edges. In a laterally finite structure, therefore, lateral standing wave resonances can arise. A lateral standing wave resonance arises when the lateral dimension W of the finite structure equals an integer number of half-wavelengths λx of the lateral propagation: W=Nλx/2. The integer N implies the order of the resonance. In the first order resonance, there is one half-wavelength within the lateral length of the structure.
Lateral standing wave resonances can arise for any thickness vibration mode (e.g., TE mode or TS mode).
In FIG. 2, the first two lateral standing wave resonances in a laterally finite plate with width W and thickness t are shown for the TE1 and the TS2 thickness vibration modes (plate wave modes). The first resonances (2a and 2c) have a lateral wavelength of λx=2W are symmetric in the width of the plate while the second resonances have a lateral wavelength λx=W (2b and 2d) are antisymmetric in the width of the plate.
Dispersion Diagrams
Relation between the lateral wave number kx of a laterally propagating plate wave and the frequency f is called the dispersion of the plate wave and is presented as a dispersion diagram. In a dispersion diagram, negative x-axis often corresponds to imaginary wave number (evanescent wave), whereas positive x-axis corresponds to real wave number (propagating wave).
In FIG. 3, a calculated dispersion diagram is shown. Second-order thickness shear (TS2) and first-order thickness extensional (TE1) plate wave modes are denoted.
Electrical Coupling of Resonance Modes
Mechanical vibration in a piezoelectric film produces an electrical field. For this field to create voltage between the electrodes of a resonator, it needs to be such that the total charge over the electrode is not zero.
Acoustical Coupling
Mechanical vibration can couple mechanically from one resonator structure to another. Mechanical coupling can happen via an evanescent wave or via a propagating wave.